Question

Prove $\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\text{csc}A+\cot A$, using the identity ${\text{csc}}^{2}A=1+{\cot }^{2}A$

Answer 1

$\frac{(\cos A-\sin A+1)}{(\cos A+\sin A-1)}$.

Divide both numerator and the denominator by $\sin A$.the sum becomes

$\frac{(\cot A-1+\csc A)}{(\cot A+1-\csc A)}$

$=\frac{(\csc A+\cot A-1)}{(\cot A-\csc A+1)}$

$=\frac{\csc A+\cot A-({\csc }^{2}A-{\cot }^{2}A)}{\cot A-\csc A+1}$

$=\frac{(\csc A+cotA)-(\csc A+\cot A)(\csc A-\cot A)}{\cot A-\csc A+1}$

$=\frac{(\csc A+cotA)[1-(\csc A-\cot A)]}{\cot A-\csc A+1}$

$=\frac{(\csc A+cotA)(1-\csc A+\cot A)}{\cot A-\csc A+1}$

$=\frac{(\csc A+\cot A)(\cot A-\csc A+1)}{\cot A-\csc A+1}$

$=(\csc A+\cot A).\frac{\cot A-\csc A+1}{\cot A-\csc A+1}$

$=\csc A+\cot A$.

Identities used:

$1+co{t}^{2}A={\csc }^{2}A$

$\csc A=\frac{1}{\sin A}$